Ch 2.5: Autonomous Equations and Population Dynamics

Transcription

1 Ch 2.5: Autonomous Equations and Population Dnamics In this section we examine equations of the form d/dt = f (), called autonomous equations, where the independent variable t does not appear explicitl. The main purpose of this section is to learn how geometric methods can be used to obtain qualitative information directl from differential equation without solving it. Example (Exponential Growth): Solution: d r, r dt e rt

2 Lemmings Suicide Mth: Dr. arl S. ruszelnicki in ABC Science Lemmings belong to the rodents. Rodents have been around for about 57 million ears. The True Lemming is about 1 cm long, with short legs and tail. Man of the rodents have strange population explosions. One such event in the Central Valle of California in had mouse populations reaching around 2, per hectare (about 2 mice per square metre). In France between 179 and 1935, there were at least 2 mouse plagues. But lemmings have the most regular fluctuations - these population explosions happen ever three or four ears. The numbers rocket up, and then drop almost to extinction. Even after three-quarters of a centur of intensive research, we don't full understand wh their populations fluctuate so much. Various factors (change in food availabilit, climate, densit of predators, stress of overcrowding, infectious diseases, snow conditions, sunspots, etc) have all been put forward, but none completel explain what is going on. Indeed, this mth is now a metaphor for the behaviour of crowds of people who foolishl follow each other, lemming-like, regardless of the consequences. This particular mth began with a Disne movie. The mth of mass lemming suicide began when the Walt Disne movie, Wild Wilderness was released in It was filmed in Alberta, Canada, far from the sea and not a native home to lemmings. So the filmmakers imported lemmings, b buing them from Inuit children. Lemmings do have their regular wild fluctuations in population and when the numbers are high, the lemmings do migrate. How do we set up models for fluctuations in population?

3 Logistic Growth An exponential model ' = r, with solution = e rt, predicts unlimited growth, with rate r > independent of population. Assuming instead that growth rate depends on population size, replace r b a function h() to obtain d/dt = h(). We want to choose growth rate h() so that h() r when is small, h() decreases as grows larger, and h() < when is sufficientl large, (i.e. Population decreases) The simplest such function is h() = r a, where a >. Our differential equation then becomes d dt r a, ra, This equation is known as the Verhulst, or logistic, equation.

4 Logistic Equation The logistic equation from the previous slide is d r a, ra, dt This equation is often rewritten in the equivalent form where = r/a. The constant r is called the intrinsic growth rate, and as we will see, represents the carring capacit of the population. A direction field for the logistic equation with r = 1 and = 1 is given here.

5 Logistic Equation: Equilibrium Solutions Our logistic equation is d dt r1, r, Two equilibrium solutions are clearl present: 1 t), ( t) ( 2 In direction field below, with r = 1, = 1, note behavior of solutions near equilibrium solutions: = is unstable, = 1 is asmptoticall stable.

6 Autonomous Equations: Equilibrium Solutions Equilibrium solutions of a general first order autonomous equation ' = f () can be found b locating roots of f () =. These roots of f () are called critical points. For example, the critical points of the logistic equation d dt r 1 are = and =. Thus critical points are constant functions (equilibrium solutions) in this setting.

7 Logistic Equation: Qualitative Analsis and Curve Sketching (1 of 7) To better understand the nature of solutions to autonomous equations, we start b graphing f () vs. In the case of logistic growth, that means graphing the following function and analzing its graph using calculus. r f ( ) r 1 ( )

8 Logistic Equation: Critical Points (2 of 7) The intercepts of f occur at = and =, corresponding to the critical points of logistic equation. The vertex of the parabola is (/2, r/4), as shown below. f ( ) r1 f ( ) r f 2 r r set r 4 2

9 Logistic Solution: Increasing, Decreasing (3 of 7) Note d/dt > for < <, so is an increasing function of t there (indicate with right arrows along -axis on < < ). Similarl, is a decreasing function of t for > (indicate with left arrows along -axis on > ). In this context the -axis is often called the phase line. d dt r1, r

10 Logistic Solution: Steepness, Flatness (4 of 7) Note d/dt when or, so is relativel flat there, and gets steep as moves awa from or. d dt r 1

11 Logistic Solution: Concavit (5 of 7) Next, to examine concavit of (t), we find '': d d 2 d 2 dt dt dt f ( ) f ( ) f ( ) f ( ) Thus the graph of is concave up when f and f ' have same sign, which occurs when < < /2 and >. The graph of is concave down when f and f ' have opposite signs, which occurs when /2 < <. Inflection point occurs at intersection of and line = /2.

12 (Example 1) Consider the logistic equation: = 1 (1) Find equilibrium solutions (2) Find a general solution of the ODE Hint: Use partial fractions

13 Solving the Logistic Equation (1 of 3) Provided and, we can rewrite the logistic ODE: d 1 rdt Expanding the left side using partial fractions, 1 A B 1 A B1 B 1, A Thus the logistic equation can be rewritten as 1 1/ d rdt 1 / Integrating the above result, we obtain ln ln1 rt C

14 Solving the Logistic Equation (2 of 3) We have: ln If < <, then < < and hence ln ln1 ln 1 rt C rt C Rewriting, using properties of logs: ln 1 rt C 1 e rtc 1 ce rt or, where () rt e

15 Solution of the Logistic Equation (3 of 3) We have: rt e for < <. It can be shown that solution is also valid for >. Also, this solution contains equilibrium solutions = and =. Hence solution to logistic equation is rt e

16 Logistic Solution: Asmptotic Behavior The solution to logistic ODE is rt e We use limits to confirm asmptotic behavior of solution: lim lim t t lim rt e t Thus we can conclude that the equilibrium solution (t) = is asmptoticall stable, while equilibrium solution (t) = is unstable. The onl wa to guarantee solution remains near zero is to make =.

17 Logistic Solution: Curve Sketching (6 of 7) Combining the information on the previous slides, we have: Graph of increasing when < <. Graph of decreasing when >. Slope of approximatel zero when or. Graph of concave up when < < /2 and >. Graph of concave down when /2 < <. Inflection point when = /2. Using this information, we can sketch solution curves for different initial conditions.

18 Logistic Solution: Discussion (7 of 7) Using onl the information present in the differential equation and without solving it, we obtained qualitative information about the solution. For example, we know where the graph of is the steepest, and hence where changes most rapidl. Also, tends asmptoticall to the line =, for large t. The value of is known as the carring capacit, or saturation level, for the species. Note how solution behavior differs from that of exponential equation, and thus the decisive effect of nonlinear term in logistic equation.

19 Example 2: Pacific Halibut (1 of 2) Let the halibut population in a region satisfies the logistic equation with r =.71/ear, = 8.5 x 1 6 kg and =.25. Let be biomass (in kg) of halibut population at time t. (a) Set up an ODE and solve the equation (b) Find the population 2 ears later (c) Find the time such that () =.75.

20 rt e Ex 2: Pacific Halibut (1 of 2) Let be biomass (in kg) of halibut population at time t, with r =.71/ear and = 8.5 x 1 6 kg. If =.25, find (a) biomass 2 ears later (b) the time such that () =.75. (a) For convenience, scale equation: Then and hence (2) rt 1 e e (.71)( 2) (2) kg

21 Example 1: Pacific Halibut, Part (b) (2 of 2) (b) Find time for which () = ears ln e e e e r r r r rt e 1

22 Critical Threshold Equation (1 of 2) Consider the following modification of the logistic ODE: d dt r1, r T The graph of the right hand side f () is given below.

23 Critical Threshold Equation: Qualitative Analsis and Solution (2 of 2) Performing an analsis similar to that of the logistic case, we obtain a graph of solution curves shown below. T is a threshold value for, in that population dies off or grows unbounded, depending on which side of T the initial value is. See also laminar flow discussion in text. It can be shown that the solution to the threshold equation is d dt r1, r T T T e rt

24 Logistic Growth with a Threshold (1 of 2) In order to avoid unbounded growth for > T as in previous setting, consider the following modification of the logistic equation: d dt r 1 1, r and T T The graph of the right hand side f () is given below.

25 Logistic Growth with a Threshold (2 of 2) Performing an analsis similar to that of the logistic case, we obtain a graph of solution curves shown below right. T is threshold value for, in that population dies off or grows towards, depending on which side of T is. is the carring capacit level. Note: = and = are stable equilibrium solutions, and = T is an unstable equilibrium solution.